This thesis is an account of our study of two branches of dynamical systems theory, namely the mean and pointwise ergodic theory. In our work on mean ergodic theorems, we investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces which includes Fr echet spaces. We show how these theorems simplify the derivation of Mean Ergodic theorems. Next we turn to the topic of pointwise ergodic theorems. We analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of -compact locally compact Hausdor groups acting measure-preservingly on - nite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on di erent Banach function spaces, and how the properties of these function spaces in- uence the weak type inequalities that can be obtained. Finally, we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages. Our investigations of these two parts of ergodic theory are uni ed by the techniques used - locally convex vector spaces, harmonic analysis, measure theory - and by the strong interaction of the nal results, which are obtained in greater generality than hitherto achieved. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2014

This thesis is an account of our study of two branches of dynamical systems theory, namely the mean and pointwise ergodic theory. In our work on mean ergodic theorems, we investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces which includes Fr echet spaces. We show how these theorems simplify the derivation of Mean Ergodic theorems. Next we turn to the topic of pointwise ergodic theorems. We analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of -compact locally compact Hausdor groups acting measure-preservingly on - nite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on di erent Banach function spaces, and how the properties of these function spaces in- uence the weak type inequalities that can be obtained. Finally, we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages. Our investigations of these two parts of ergodic theory are uni ed by the techniques used - locally convex vector spaces, harmonic analysis, measure theory - and by the strong interaction of the nal results, which are obtained in greater generality than hitherto achieved. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2014

Dans cette thèse, nous prouvons des résultats de théorie spectrale, directe et inverse, dans la limite semi-classique, pour les opérateurs de Toeplitz autoadjoints sur les surfaces. Pour les opérateurs pseudo-différentiels, les résultats en question sont déjà connus, et il est naturel de vouloir les étendre aux opérateurs de Toeplitz. Les conditions de Bohr-Sommerfeld usuelles, qui caractérisent les valeurs propres proches d'une valeur régulière du symbole principal, ont été obtenues il y a quelques années seulement pour les opérateurs de Toeplitz. Notre contribution consiste en l'extension de ces conditions près de valeurs critiques non dégénérées. Nous traitons le cas d'une valeur critique elliptique à l'aide d'une technique de forme normale ; l'opérateur modèle est la réalisation de l'oscillateur harmonique sur l'espace de Bargmann, dont le spectre est bien connu. Dans le cas d'une valeur critique hyperbolique, la forme normale ne suffit plus et nous complétons l'étude en faisant appel à des arguments dus à Colin de Verdière et Parisse, à qui l'on doit le résultat analogue dans le cas pseudo-différentiel. Enfin, nous établissons un résultat de théorie spectrale inverse pour les opérateurs de Toeplitz autoadjoints sur les surfaces ; plus précisément, nous montrons que sous certaines hypothèses génériques, la connaissance du spectre à l'ordre deux dans la limite semi-classique permet de retrouver le symbole principal à symplectomorphisme près. Ce résultat s'appuie en grande partie sur l'écriture des règles de Bohr-Sommerfeld. / In this thesis, we prove some direct and inverse spectral results, in the semiclassical limit, for self-adjoint Toeplitz operators on surfaces. For pseudodifferential operators, these results are already known, and it is natural to expect their extension to the Toeplitz setting. The usual Bohr-Sommerfeld conditions, characterizing the eigenvalues close to a regular value of the principal symbol, have been obtained a few years ago for Toeplitz operators. Our contribution consists in extending these conditions near nondegenerate critical values. We handle the case of an elliptic value thanks to a normal form technique; the model operator is the realization of the harmonic oscillator in the Bargmann space, whose spectrum is well-known. In the case of a hyperbolic value, the normal form is no longer sufficient and we conclude by using additional arguments due to Colin de Verdière and Parisse, who derived the analogous result for pseudodifferential operators. Finally, we write an inverse spectral result for self-adjoint Toeplitz operators on surfaces; more precisely, we show that under some generic hypotheses, the knowledge of the spectrum up to order two in the semiclassical limit allows to recover the principal symbol up to symplectomorphism. This result essentially relies on Bohr-Sommerfeld rules.

Ce travail porte sur le problème des partitions minimales, à l'interface entre théorie spectrale et optimisation de forme. Une introduction générale précise le problème et présente des résultats, principalement dûs à B. Helffer, T. Hoffmann-Ostenhof et S. Terracini, qui sont utilisés dans le reste de la thèse.Le premier chapitre est une étude spectrale asymptotique du laplacien de Dirichlet sur une famille de domaines en dimension deux qui tend vers un segment. L'objectif est d'obtenir une localisation des lignes nodales dans la limite des domaines minces. En appliquant les résultats de Helffer, Hoffmann-Ostenhof et Terracini, on montre ainsi que les domaines nodaux des premières fonctions propres forment des partitions minimales.Le deuxième chapitre étudie les valeurs propres de certains opérateurs de Schrödinger sur un domaine plan avec condition au bord de Dirichlet. On considère des opérateurs qui ont un potentiel électrique nul et un potentiel magnétique d'un type particulier, dit d'Aharonov-Bohm, avec des singularités en un nombre fini de points appelés pôles. On démontre que les valeurs propres dépendent continuement des pôles. Dans le cas de pôles distincts et éloignés du bord, on prouve que cette dépendance est analytique lorsque la valeur propre est simple. On exprime de plus une condition suffisante pour que la fonction qui aux pôles associe une valeur propre présente un point critique. On utilise alors la caractérisation magnétique des partitions minimales pour montrer que l'énergie minimale est une valeur critique d'une de ces fonctions.Le troisième chapitre est un article écrit en collaboration avec Virginie Bonnaillie-Noël. Il porte sur une famille d'exemples, les secteurs angulaires de rayon unité et d'ouverture variable, dont on tente de déterminer les partitions minimales. On applique pour cela les théorèmes généraux rappelés dans l'introduction afin de déterminer les partitions nodales qui sont minimales. On s'intéresse plus particulièrement aux partitions minimales en trois domaines. En appliquant les idées du deuxième chapitre, on montre que pour certaines valeur de l'angle, il n'existe aucune partition minimale qui soit symétrique par rapport à la bissectrice du domaine. D'un point de vue quantitatif, on obtient des encadrements précis de l'énergie minimale.Le quatrième chapitre consiste en l'étude des partitions minimales de tores plats dont on fait varier le rapport entre longueur et largeur. On utilise une méthode numérique très différente de celle du troisième chapitre, basée sur un article de B. Bourdin, D. Bucur et É. Oudet. Elle consiste en une relaxation suivie d'une optimisation par un algorithme de gradient projeté. On peut ainsi tester des résultats théoriques antérieurs. Les résultats présentés suggèrent de plus la construction explicite de familles de partitions (en liaison avec des pavages du tore) qui donnent une nouvelle majoration de l'énergie minimale.Un dernier chapitre de perspectives présente plusieurs applications possibles des méthodes décrites dans la thèse. / This work is concerned with the problem of minimal partitions, at the interface between spectral theory and shape optimization. A general introduction gives a precise statement of the problem and recall results, mainly due to B. Helffer, T. Hoffmann-Ostenhof and S.Terracini, that are used in the rest of the thesis.The first chapter is an asymptotic spectral study of the Dirichlet Laplacian on a familly of two-dimensional domains converging to a line segment. The aim is to localize the nodal lines when the domains become very thin. With the help of the results of Helffer, Hoffmann-Ostenhof, and Terracini, we then show that the nodal domains of the first eigenfunctions give minimal partitions.The second chapter studies the eigenvalues of some Schrödinger operators on a domain with Dirichlet boundary conditions. We consider operators that have no electric potential and a so-called Aharonov-Bohm magnetic potential, which has singularities at a finite number of points called poles. We prove that the eigenvalues are continuous functions of the poles. When the poles are distinct and far from the boundary, we prove that this function is analytic, assuming the eigenvalue is simple. We also give a sufficient condition for the function to have a critical point. Using the magnetic characterization of minimal partitions, we show that the minimal enery is a critical value for one of these functions.The third chapter in an article written in collaboration with Virginie Bonnaillie-Noël. It studies minimal partitions for sectors of unit radius with a variable angular opening. We apply the general results presented in the introduction, together with numerical computations, to determine nodal partitions that are minimal. We focus on partitions into three domains. Using ideas from the second chapter, we show that, for some values of the angle, there is no minimal partition that is symmetric with respect to the bisector. Form a quantitative point of view, we obtain precise bounds on the minimal energy.The fourth chapter studies the minimal partitions of flat tori in function of the ratio between width and length. We use a numerical method that is quite different from chapter three, and is based on an article by B. Bourdin, D. Bucur, and É. Oudet. It consists in a relaxation of the problem, followed by optimization with the help of a projected gradient algorithm. The results shown here additionally suggest explicit families of partitions, which consist in tilings of tori by polygons, that give upper bounds on the minimal energy. In the last chapter we consider several possible applications of the methods described in the thesis.

La thèse de doctorat comporte trois parties rédigées en anglais. Les deux premières parties correspondent principalement à l'étude de l'énergie de l'état fondamental. La dernière partie est consacrée à l'analyse de l'effet de pinning dans la supraconductivité.Dans une première partie de cette thèse, nous considérons la fonctionnelle de Ginzburg -Landau avec un champ magnétique variable appliqué dans un domaine borné et régulier de dimension 2. Nous déterminons le comportement asymptotique du paramètre d'ordre dans le régime o\`u le paramètre de Ginzburg-Landau et le champ magnétique sont grands et de même ordre. Comme conséquence, nous montrons que le paramètre d'ordre est localisé asymptotiquement dans la région où le profil du champ magnétique appliqué est petit.Dans une autre partie, nous considérons la fonctionnelle de Ginzburg -Landau avec un champ magnétique variable appliqué dans un domaine borné et régulier de dimension 2. Le profil du champ magnétique appliqué varie régulièrement et peut s'annuler exactement à l'ordre 1 le long d'une courbe. En supposant que la l'intensité du champ magnétique appliqué varie entre deux échelles caractéristiques, et que le paramètre de Ginzburg- Landau tend vers l'infini, nous déterminons une formule asymptotique précise pour minimiser l'énergie et montrer que les minimiseurs de l'énergie ont des vortex. Nous mettons en évidence que la présence d'un champ magnétique variable implique que la distribution de la vorticité dans l'échantillon n'est pas uniforme.Dans la dernière partie, nous étudions l'énergie de Ginzburg-Landau d'un supraconducteur avec un champ magnétique variable et un terme de pinning dans un domaine borné et régulier de dimension 2. En supposant que le paramètre de Ginzburg-Landau et l'intensité du champ magnétique sont grands et de même ordre, nous déterminons une formule asymptotique précise pour l'énergie. De plus, nous discutons l'existence des solutions non-triviales et déterminons le comportement asymptotique du troisième champ critique de la supraconductivité. / The PHD thesis has three parts, the first and the second part correpond mainly to study the groundstate energy, the last one being devoted to the analysis of the pinning effect in superconductivity.In a first part of this thesis, we consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two-dimensional domain. We determine an accurate asymptotic formula for the minimizing energy when the Ginzburg-Landau parameter and the magnetic field are large and of the same order. As a consequence, it is shown how bulk superconductivity decreases in average as the applied magnetic field increases.In another part, we consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two-dimensional domain. The profile of the applied magnetic field varies smoothly and is allowed to vanish non-degenerately along a curve. Assuming that the strength of the applied magnetic field varies between two characteristic scales, and that the Ginzburg-Landau parameter tends to , we determine an accurate asymptotic formula for the minimizing energy and show that the energy minimizers have vortices. The new aspect in the presence of variable magnetic field is that the distribution of vortices in the sample is not uniform.In the final part, we study the Ginzburg-Landau energy of a superconductor with a variable magnetic field and a pinning term in a bounded and smooth two-dimensional domain . Supposing that the Ginzburg-Landau parameter and the intensity of magnetic field are large and of the same order, we determine an accurate asymptotic formula for the minimizing energy. Also, we discuss the existence of non-trivial solutions and prove an asymptotics of the third critical field.

Neste trabalho, consideramos o problema autônomo {(-∆u+V(x)u=f(u) em R^N,@u∈H^1 (R^N)\\{0},)┤ em que N≥3, a função V é não periódica, radialmente simétrica e muda de sinal e a não linearidade f é assintoticamente linear. Além disso, impomos que V possui um limite positivo no infinito e que o espectro do operador L≔-∆+V tem ínfimo negativo. Sob essas condições, baseando-se em interações entre soluções transladadas do problema no infinito associado, é possível mostrar que tal problema satisfaz a geometria do teorema de linking clássico e garantir a existência de uma solução fraca não trivial. Em seguida, estabelecemos a existência de uma solução não trivial para o problema não autônomo {(-∆u+V(x)u=f(x,u) em R^N,@u∈H^1 (R^N)\\{0},)┤ sob hipóteses similares ao problema anterior, admitindo também que f(x,u)=f(|x|,u) dentre outras condições. Aplicamos novamente o teorema de linking para garantir que tal problema possui uma solução não trivial. Por fim, provamos que o problema quasilinear {(-∆u+V(x)u-u∆(u^2)=g(x,u) em R^3,@u∈H^1 (R^3)\\{0},)┤ em que o potencial V muda de sinal, podendo ser não limitado inferiormente, e a não linearidade g(x,u), quando |x|→∞, possui um certo tipo de monotonicidade, possui uma solução não trivial. A existência de tal solução é provada por meio de uma mudança de variável que transforma o problema num problema semilinear, nos permitindo, assim, empregar o teorema do passo da montanha combinado com o lema splitting. / In this work, we consider the autonomous problem {(-∆u+V(x)u=f(u) em R^N,@u∈H^1 (R^N)\\{0},)┤ where N≥3, V is a non-periodic radially symmetric function that changes sign and the nonlinearity f is asymptotically linear. Furthermore, we impose that V has a positive limit at infinity and the spectrum of the operator L≔-∆+V has negative infimum. Under these conditions, employing interaction between translated solutions of the problem at infinity, it is possible to show that such problem satisfies the geometry of the classical linking theorem and garantee the existence of a nontrivial weak solution. After that, we establish the existence of a nontrivial weak solution for the nonautonomous problem {(-∆u+V(x)u=f(x,u) em R^N,@u∈H^1 (R^N)\\{0},)┤ under similar hyphoteses to the previous problem, assuming also that f(x,u)=f(|x|,u) among others conditions. We apply again the classical linking theorem to ensure that such problem possesses a nontrivial weak solution. Finally, we prove that the quasilinear problem {(-∆u+V(x)u-u∆(u^2)=g(x,u) em R^3,@u∈H^1 (R^3)\\{0},)┤ where the potential V changes sign and may be unbounded from below and the nonlinearity g(x,u), as|x|→∞, has a kind of monotonicity, has a nontrivial weak solution. The existence of such solution is proved by means of a change of variables that makes the problem become a semilinear problem and hence allow us apply the mountain pass theorem combined with splitting lemma.

<p>The topic of this thesis is computation of Mass waveforms, and we consider a number of different cases: Congruence subgroups of the modular group and Dirichlet characters (chapter 1); congruence subgroups and general multiplier systems and real weight (chapter 2); and noncongruence subgroups (chapter 3). In each case we first discuss the necessary theoretical background. We then outline the algorithm and display some of the results obtained by it.</p>

The topic of this thesis is computation of Mass waveforms, and we consider a number of different cases: Congruence subgroups of the modular group and Dirichlet characters (chapter 1); congruence subgroups and general multiplier systems and real weight (chapter 2); and noncongruence subgroups (chapter 3). In each case we first discuss the necessary theoretical background. We then outline the algorithm and display some of the results obtained by it.

The work of this Ph.D. thesis in mathematics concerns the problem of determining existence, uniqueness, and structure of zero-energy states in supersymmetric matrix models, which arise from a quantum mechanical description of the physics of relativistic membranes, reduced Yang-Mills gauge theory, and of nonperturbative features of string theory, respectively M-theory. Several new approaches to this problem are introduced and considered in the course of seven scientific papers, including: construction by recursive methods (Papers A and D), deformations and alternative models (Papers B and C), averaging with respect to symmetries (Paper E), and weighted supersymmetry and index theory (Papers F and G). The mathematical tools used and developed for these approaches include Clifford algebras and associated representation theory, structure of supersymmetric quantum mechanics, as well as spectral theory of (matrix-) Schrödinger operators. / QC20100629

I denna avhandling, som omfattar fyra forskningsartiklar, betraktas två operatorer inom den matematiska fysiken. De båda tidigare artiklarna innehåller resultat för Schrödingeroperatorn med Aharonov-Bohm-magnetfält. I artikel I beräknas spektrum och egenfunktioner till denna operator i R2 explicit i ett antal fall då en radialsymmetrisk skalärvärd potential eller ett konstant magnetfält läggs till. I flera av de studerade fallen kan den skarpa konstanten i Lieb-Thirrings olikhet beräknas för γ = 0 och γ ≥ 1. I artikel II bevisas semiklassiska uppskattningar för moment av egenvärdena i begränsade tvådimensionella områden. Vidare presenteras ett exempel då den generaliserade diamagnetiska olikheten, framlagd som en förmodan av Erdős, Loss och Vougalter, är falsk. Numeriska studier kompletterar dessa resultat. De båda senare artiklarna innehåller ett flertal spektrumuppskattningar för Heisenberg-Laplace-operatorn. I artikel III bevisas skarpa olikheter för spektret till Dirichletproblemet i (2n + 1)-dimensionella områden med ändligt mått. Låt λk och μk beteckna egenvärdena till Dirichlet- respektive Neumannproblemet i ett område med ändligt mått. N. D. Filonov har bevisat olikheten μk+1 &lt; λk för den euklidiska Laplaceoperatorn. I artikel IV visas detta resultat för Heisenberg-Laplaceoperatorn i tredimensionella områden som uppfyller vissa geometriska villkor. / In this thesis, which comprises four research papers, two operators in mathe- matical physics are considered. The former two papers contain results for the Schrödinger operator with an Aharonov-Bohm magnetic field. In Paper I we explicitly compute the spectrum and eigenfunctions of this operator in R2 in a number of cases where a radial scalar potential and/or a constant magnetic field are superimposed. In some of the studied cases we calculate the sharp constants in the Lieb-Thirring inequality for γ = 0 and γ ≥ 1. In Paper II we prove semi-classical estimates on moments of the eigenvalues in bounded two-dimensional domains. We moreover present an example where the generalised diamagnetic inequality, conjectured by Erdős, Loss and Vougalter, fails. Numerical studies complement these results. The latter two papers contain several spectral estimates for the Heisenberg Laplacian. In Paper III we obtain sharp inequalities for the spectrum of the Dirichlet problem in (2n + 1)-dimensional domains of finite measure. Let λk and μk denote the eigenvalues of the Dirichlet and Neumann problems, respectively, in a domain of finite measure. N. D. Filonov has proved that the inequality μk+1 &lt; λk holds for the Euclidean Laplacian. In Paper IV we extend his result to the Heisenberg Laplacian in three-dimensional domains which fulfil certain geometric conditions. / QC 20100712